The value of `int(a x^2-b)/(xsqrt(c^2x^2-(a x^2+b)^2))` (a) `1/csin^(-1)(a x+b/x)+k` (b) `csin^(-1)(a+b/x)+e` (c) `sin^(-1)((a x+b/x)/c)+k` (d) none of these

Options

A. `(1)/(c)sin^(-1)(ax+(b)/(x))+k`
B. `c sin^(-1)(a+(b)/(x))+k`
C. `sin^(-1)((ax+(b)/(x))/(c))+k`
D. none of these

Answer

C

Explanation

Let `I=int ((ax^(2)-b)dx)/(xsqrt(c^(2)x^(2)-(ax^(2)+b)^(2)))`
`=int((a-(b)/(x^(2)))dx)/(sqrt(c^(2)-(ax+(b)/(x))^(2))), " "{("Put "ax+(b)/(x)=t),( :.(a-(b)/(x^(2)))dx=dt):}`
`=int(dt)/(sqrt(c^(2)-t^(2)))`
` =sin^(-1)((t)/(c))+k`
`=sin^(-1)((ax+(b)/(x))/(c))+C`